I'm reading about Skolem. And I'm wondering about the result of the paradox: that countability (at least in first-order formulations) is relative. Now, even when we state Skolem's theory -- if a first-order theory has an infinite model then it has a countable model -- from what "perspective" is this model countable? From another model? Absolutely?

If we say that a model M is countable "from its own perspective" that means there is some bijection (set) B in the domain of M containing every set in the domain of M (including B) and the set of natural numbers. But that can't happen, because then B would be a member of itself.

So, when we casually talk about countability, we take it to be an absolute notion. But, what does this say about Skolem's Paradox?

May 26th 2012, 09:42 AM

nomadreid

Re: Can a model say of itself that it is countable?

First of all, the phrasing is a little odd. A model does not "say" anything: a sentence in a first-order theory says something about a variable, and the sentence is satisfied or not by substitution of elements from the universe of the model for that variable. Given that the universe of the model cannot be an element of itself, the sentence cannot say anything about the universe of the model, much less of the model itself. Second-order logic will also not allow the formula to say anything about the universe, because although the variables can now range over subsets of the universe, the universe is a proper class, not a subset. So, in order to say anything about this model, you need to use a new universe which has the old universe as a subset, and the appropriate relations in the new universe. And then you can have a theory containing sentences which say that the old universe is countable.
Countability is not an absolute notion.

May 26th 2012, 12:55 PM

mpitluk

Re: Can a model say of itself that it is countable?

Quote:

Originally Posted by nomadreid

First of all, the phrasing is a little odd. A model does not "say" anything: a sentence in a first-order theory says something about a variable, and the sentence is satisfied or not by substitution of elements from the universe of the model for that variable. Given that the universe of the model cannot be an element of itself, the sentence cannot say anything about the universe of the model, much less of the model itself. Second-order logic will also not allow the formula to say anything about the universe, because although the variables can now range over subsets of the universe, the universe is a proper class, not a subset. So, in order to say anything about this model, you need to use a new universe which has the old universe as a subset, and the appropriate relations in the new universe. And then you can have a theory containing sentences which say that the old universe is countable.
Countability is not an absolute notion.

Right. There is no function in a (standard) model that takes the natural numbers to each member of the domain.

So we can say a model M is countable only when M is a subset of a different model M' and there is a bijection f in M' between N and M.

But even though M is countable, there is some A in M such that M satisfies "A is uncountable."

Hence the relativity of countability: it depends on what a model takes to be w.

I was thinking that when we say a model is countable, we say it from a naive categoricity perspective. But Lowenheim-Skolem's theorem show that there is no such perspective.

We can always question the applicability of a mathematical result to the real world. That was what I was trying to do here and in other posts. So, I was wondering about countability in the real world, outside model-theoretic statements.

This naturally brings up the following questions: what are the natural numbers in the real world, what is a bijection in the real world?

I don't know the answers, but I was wondering what others thought about it -- their personal ideas, not the parroting of other philosophers/mathematicians.

May 27th 2012, 02:45 PM

Deveno

Re: Can a model say of itself that it is countable?

i would answer that there are no natural numbers in the real world, nor no bijections. in other words mathematics models reality, it is NOT reality. some people have wondered why mathematics is so successful in this regard. i think of it like this:

ABC ~ D

where A is abstraction of "properties of reality", B is pure mathematical (logical) reasoning, C is "specification" or "instantiation" of the reasoning back to reality, and D is the actual physical world. so why this works is that B and C are "inverses" of each other, or perhaps more precisely "dual".

for example, take counting sheep. every sheep is unique, but we abstract the property "is a sheep", and perform cardinal arithmetic on "the set of sheep". we get some number k, and conclude "there are k sheep". the reality is far more complex, each sheep has a unique genetic code, a distinct appearance, position in space, etc. we "idealize" the situation. we need to be aware of the fact that when we speak about something, the ideas we express in our communication, are our own creations, they don't exist "out there". what exists "out there" is something inexpressible, and far more complicated than ANY language can express. meaning is literally created by inaccuracy (omitting irrelevant detail).

talking about "apples and oranges" and talking about "fruit" leads to two different conversations, neither of which is "wrong" and neither of which perfectly describes what we're talking about. what i think work on the foundations of mathematics has indicated, is that mathematics does not possess a single monolithic foundation, that we have to choose a (partial) perspective, and work within that. the "standard framework":

has proved to be particularly fruitful in terms of describing physical phenomenon, but that doesn't mean it is the only possible framework.

December 10th 2012, 05:07 PM

FelipeAbraham

Re: Can a model say of itself that it is countable?

Hi.

I think the best way to undertand Skolemīs "Paradox" is to prove that the function that maps the real numbers in a countable structure (which is a set) of a model of ZFC (for example) cannot belong to that structure, unless ZFC is inconsistent. One can show this mathematically. Until here we are fine.

The knot in our minds appears when we ask where is this function so.
The key step is to see that we are proving this outside of the ZFC (or any other strong set theory we use). This is Model theory talking about Set theory (in our case, ZFC), not ZFC talking about model theory that talks about the former ZFC system. This model theory we use to prove those things isnīt running on ZFC. We are using another axiomatic system.

But cannot we run Model Thery on ZFC, for example. Yes we can run Model theory on any strong set theory. But what the hell so?
Here is the beauty of the skolemīs paradox. It is ZFC talking about ZFC and never talking about itself.

Since we can never define what is true in ZFC with a formula of ZFC (Tarskiīs theorem), any formula that does it (for example, the formula saying a formula of ZFC is satisfiable by some model) cannot say everything that is true in ZFC. So, supposing ZFC consistent we prove Lowenheim-skolem theorem will never contradict Cantorīs Theorem.

It is a very captive result. And I think maybe it is deeper than Gödelīs theorem.